v2010.10.26 - Convex Optimization

E.9. PROJECTION ON CONVEX SET 747accomplished by first projecting orthogonally on that subspace, and thenprojecting the result on C ; [109,5.14] id est, the ordered product of twoindividual projections that is not commutable.Proof. (⇐) To show that, suppose unique minimum-distance projectionP C x on C ⊂ R n is y as illustrated in Figure 166;‖x − y‖ ≤ ‖x − q‖ ∀q ∈ C (2043)Further suppose P Rn x equals z . By the Pythagorean theorem‖x − y‖ 2 = ‖x − z‖ 2 + ‖z − y‖ 2 (2044)because x − z ⊥ z − y . (1907) [250,3.3] Then point y = P C x is the sameas P C z because‖z − y‖ 2 = ‖x − y‖ 2 − ‖x − z‖ 2 ≤ ‖z − q‖ 2 = ‖x − q‖ 2 − ‖x − z‖ 2which holds by assumption (2043).(⇒) Now suppose z = P Rn x and∀q ∈ C(2045)‖z − y‖ ≤ ‖z − q‖ ∀q ∈ C (2046)meaning y = P C z . Then point y is identical to P C x because‖x − y‖ 2 = ‖x − z‖ 2 + ‖z − y‖ 2 ≤ ‖x − q‖ 2 = ‖x − z‖ 2 + ‖z − q‖ 2by assumption (2046).∀q ∈ C(2047)This proof is extensible via translation argument. (E.4) Uniqueminimum-distance projection on a convex set contained in an affine subsetis, therefore, similarly accomplished.Projecting matrix H ∈ R n×n on convex cone K = S n ∩ R n×n+ in isomorphicR n2 can be accomplished, for example, by first projecting on S n and only thenprojecting the result on R n×n+ (confer7.0.1). This is because that projectionproduct is equivalent to projection on the subset of the nonnegative orthantin the symmetric matrix subspace.